If $f(x)=\left|\begin{array}{ccc}x-3 & 2 x^2-18 & 3 x^3-81 \ x-5 & 2 x^2-50 & 4 x^3-500 \ 1 & 2 & 3\end{array}\right|$, then $f(\mathrm{l}) f(3)+f(3) f(5)+f(5) f(\mathrm{l})$ is equal to

Question

If $f(x)=\left|\begin{array}{ccc}x-3 & 2 x^2-18 & 3 x^3-81 \ x-5 & 2 x^2-50 & 4 x^3-500 \ 1 & 2 & 3\end{array}\right|$, then $f(\mathrm{l}) f(3)+f(3) f(5)+f(5) f(\mathrm{l})$ is equal to, $f(1)$, $f(3)$, $f(1)+f(3)$, $f(1)+f(5)$

MARKS App · Accepted Answer

$f(x)=(x-3)(x-5)\left|\begin{array}{ccc}1 & 2(x+3) & 3\left(x^2+9+3 x\right) \ 1 & 2(x+5) & 4\left(x^2+25+5 x\right) \ 1 & 2 & 3\end{array}\right|$ $\Rightarrow \quad f(3)=f(5)=0$ So, $f(1) f(3)+f(3) \cdot f(5)+f(5) f(1)=0=f(3)$

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